Consider the right triangle shown below where a=8.09, b=9.4, and c=12.4. Note that θ and ϕ are measured in radians.What is the value of cos(θ)?cos(θ)= What is the value of sin(θ)?sin(θ)=What is the value of tan(θ)?tan(θ)= What is the value of θ?θ=
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By definition
[tex]\cos (angle)=\frac{\text{ adjacent side}}{\text{ hipotenuse}}[/tex]From the picture
[tex]\begin{gathered} \cos (\theta)=\frac{a}{c} \\ \cos (\theta)=\frac{8.09}{12.4} \\ \cos (\theta)=0.65 \end{gathered}[/tex]By definition
[tex]\sin (angle)=\frac{\text{ opposite side}}{\text{ hipotenuse}}[/tex]From the picture:
[tex]\begin{gathered} \sin (\theta)=\frac{b}{c} \\ \sin (\theta)=\frac{9.4}{12.4} \\ \sin (\theta)=0.76 \end{gathered}[/tex]By definition
[tex]\tan (angle)=\frac{\text{ opposite side}}{\text{ adjacent side}}[/tex]From the picture
[tex]\begin{gathered} \tan (\theta)=\frac{b}{a} \\ \tan (\theta)=\frac{9.4}{8.09} \\ \tan (\theta)=1.16 \end{gathered}[/tex]Isolating θ from the previous equations:
[tex]\begin{gathered} \theta=\arccos (0.65)=49.46\text{ \degree}\approx49\text{ \degree} \\ \theta=\arcsin (0.76)=49.46\text{ \degree}\approx49\text{ \degree} \\ \theta=\arctan (1.16)=49.24\text{ \degree}\approx49\text{ \degree} \end{gathered}[/tex](The difference between the values is caused by rounding errors)
Related Questions
Given the special right triangle, find the value of x and y. Express your answer in simplest radical form.
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Find x.special 10A. 3B. 23√3- this is in fractionC. 6√3D. 3√3
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First, we need to remember the cosine formula which is: cosine(theta)= adjacent/hypotenuse, now let's apply the formula to the triangle we have:
By using the formula we find that x=3√3 .
The answer is D.
A ball bounces to a height of 6.1 feet on the first bounce. Each subsequent bounce reaches a height that is 82% of the previous bounce. What is the height, in feet, of the fifth bounce? Round your answer to the thousandths place.
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In the first bounce, the height is
[tex]6.1\times(0.82)^0=6.1[/tex]In the second bounce, the height is
[tex]6.1\times(0.82)^2=5.002[/tex]Then, we can note that the pattern is
[tex]6.1\times(0.82)^{n-1}[/tex]where n represents the number of bounces of the ball. Then, for n=5 (fifth bounce), we get
[tex]\begin{gathered} 6.1\times(0.82)^{5-1} \\ 6.1\times(0.82)^4 \end{gathered}[/tex]which gives
[tex]6.1\times(0.82)^4=2.7579[/tex]Therefore, by rounding to the nearest thousandths, the answer is 2.758 feet
There were eight questions on Emily's math quiz, and she missed two questions.Which of the following diagrams represents the percentage of Emily's accuracy onthe quiz?A. 50%B. 75%C. 30%D. 10%
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There are eight quartion in the quiz and two question missed. So Emily solved six question of the quiz.
Determine the accuracy of Emily.
[tex]\frac{6}{8}\times100=75[/tex]So Emily's accuracy is 75% and option B is correct.
Describe the features of the function that can be easily seen when a quadratic function is givenin the form: y = ax2 + bx + c and how they can be identified from the equation. How can thisform be used to find the other features of the graph?
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Hello there. To solve this question, we need to remember some properties about quadratic functions and its key features.
Let f(x) = ax² + bx + c, for a not equal to zero.
The main key feature we can see at first glance is the leading coefficient a.
If a < 0, the parabola (the graph of the function) will have its concavity facing down.
If a > 0, the parabola will have its concavity facing up.
It also means the function will have either a maximum or a minimum point on its vertex, respectively.
Another key feature of the function is the y-intercept, i. e. the point in which the x-coordinate is equal to zero, is (0, c).
The x-intercepts of the graph (in plural), are the roots of the function.
If b² - 4ac > 0, we'll have two distinct real roots.
If b² - 4ac = 0, we'll have two equal real roots.
If b² - 4ac < 0, we'll have two conjugate complex roots (not real roots)
This b² - 4ac is the discriminant of the function.
The roots can be found by the formula:
x = (-b +- sqrt(b² - 4ac))/2a
The vertex of the graph can be found on the coordinates (xv, yv), in which xv is calculated by the arithmetic mean of the roots
xv = ((-b + sqrt(b²-4ac))/2a + (-b-sqrt(b²-4ac))/2a)/2 = -b/2a
The yv coordinate can be found by plugging in xv in the function
yv = a(-b/2a)² + b(-b/2a) + c, which will be equal to -(b²-4ac)/4a.
Select the statement that accurately describes the following pair oftriangles.
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In any pair of similar triangles, (side side side )
Each correspondent side has the same ratio so let's examine
ΔCDE and ΔFGH
Hello. I would like help with problem. Quick answer is OK.Thank you
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not continuous, 2 holes. Option A is correct
Explanations;For a function to be continuous, the left hand limit of a function must be equal to the right hand limit at the point x = a
From the graph shown you can see that the limit of the function from the left is not equal to the limit of the function from the right at x = 0. Therefore, we can conclude that there are discontinuities at x = 0.
You can also see that the function has 2 holes at (0, 0) and (0, -1).
what is the area of the triangle below with a side length of 4
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All angles of triangles is equal means that that triangle is equailateral triangle with side of a = 4 in.
The formula for the area of equilateral triangle is,
[tex]A=\frac{\sqrt[]{3}a^2}{4}[/tex]Substitute 4 for a in the formula to determine the area of the triangle.
[tex]\begin{gathered} A=\frac{\sqrt[]{3}}{4}\cdot(4)^2 \\ =4\sqrt[]{3} \end{gathered}[/tex]So area of triangle is,
[tex]4\sqrt[]{3}[/tex]URGENT!!! help!!!!!!!!!!!!!
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Triangles' resemblance is reflected by their congruence. If the matching sides and angles of two triangles match, the triangles are said to be congruent.
For triangles, there are five primary congruency rules: Side-Side-Side is an SSS criterion. The side-angle-side SAS criterion. Angle, Side, Angle is an ASA criterion. Angle-Angle-Side is an AAS criterion.
The midpoint of a line segment is known as the midpoint in geometry. It is the centroid of the segment and of the ends, and it is equally distant from both of them. It cuts the section in half.
An isosceles triangle in geometry is one with at least two equal-length sides. It is sometimes stated as having exactly two equal-length sides and other times as having at least two equal-length sides, with the latter version adding the equilateral triangle as one of the possible configurations.
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Could I assistance receive some on this question it’s very confusing
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We need to translate the vertex F of triangle BDF. When we translate it 2 units to the left and 4 units down, we obtain the point F'.
We know that triangle BDF has vertices B(4,3), D(6,3), and F(6,1).
The first coordinate of each point represents its x-coordinate (the distance from the y-axis). And the second coordinate of each point represents its y-coordinate (the distance from the x-axis).
So, this triangle is shown below:
Now, we need to translate the point F 2 units to the left, to obtain the redpoint below. And then translate it 4 units down, to obtain F' (the yellow point):
Therefore, the F' has coordinates:
F'(4,-3)
The quadratic equation y= -16t^2 +4t+2 represents a moving objects trajectory where y is the objects height in feet above the ground after t seconds . At what time will the objects hit the ground ?
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Since y is the object's height, it will be on the ground when y = 0. So let's do that:
[tex]0=-16t^2+4t+2[/tex]Here, we can use Bhaskara's Formula to find the roots of the equation:
[tex]\begin{gathered} t=\frac{-4\pm\sqrt[]{4^2-4\cdot(-16)\cdot2}}{2\cdot(-16)} \\ t=\frac{-4\pm\sqrt[]{16+128}}{-32}=\frac{-4\pm\sqrt[]{144}}{-32}=\frac{-4\pm12}{-32} \\ t_1=\frac{-4+12}{-32}=\frac{8}{-32}=-0.25 \\ t_2=\frac{-4-12}{-32}=\frac{-16}{-32}=0.5 \end{gathered}[/tex]Since the time at start is 0, we can't have a negative sign, it would be like saying what happened before the object was in the air. The it will hit the ground at t = 0.5 s.
Is the slope the same or different?Is the Y-intercept same or different?Is there infinitely many solutions or not?
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Answer:
Explanation:
Here, we want to answer the questions given
a) To answer this, we have to write the equations in the slope-intercept form:
The slope-intercept form is:
[tex]y\text{ = mx + b}[/tex]m is the slope while b is the y-intercept
The equations would be:
[tex]\begin{gathered} y\text{ = 7x-2} \\ y\text{ = 7x-2} \end{gathered}[/tex]We can see that the equations are same
Since the equations are same, the slope is same which is 7
b) The y-intercept value is same too
c) Since the equations are same, there are infinitely many solutions for the system of equations
At a company meeting, there were 40 people in attendance. 70% of them were managers. How many managers were in the meeting?
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triangle XZW ~ triangle XYV, find the perimeter of triangle XZW
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176.4
Explanation
as the triangle are similar we can set a proportion
Step 1
find the YZ value
a) let
[tex]ratio1=\frac{hypotenuse}{rigth\text{ side}}[/tex]so,for triangle XZW
[tex]ratio=\frac{40+32}{28+YZ}[/tex]and for triangle XYV
[tex]ratio=\frac{40}{28}[/tex]as the ratios are equal, we can set a proportion
[tex]\frac{40+32}{28+YZ}=\frac{40}{28}[/tex]b) now,solve for YZ
[tex]\begin{gathered} \frac{40+32}{28+YZ}=\frac{40}{28} \\ \frac{72}{28+YZ}=\frac{40}{28} \\ cross\text{ multiply} \\ 72*28=40(28+YZ) \\ 2016=1120+40YZ \\ subtract\text{ 1120 in both sides} \\ 2016-1120=1120+40YZ-1120 \\ 896=40YZ \\ divide\text{ bothsides by 40} \\ \frac{896}{40}=\frac{40YZ}{40} \\ 22.4=YZ \end{gathered}[/tex]so
YZ=22.4
Step 2
find the length of the side WZ
a) let
[tex]ratio=\frac{hypotenuse\text{ }}{base}[/tex]hence
[tex]\begin{gathered} ratio_1=\frac{40+32}{WZ}=\frac{72}{WZ} \\ ratio_2=\frac{40}{30} \end{gathered}[/tex]set the proportion and solve for YZ
[tex]\begin{gathered} ratio_1=\text{ ratio}_2 \\ \frac{72}{WZ}=\frac{40}{30} \\ cross\text{ multiply} \\ 72*30=40WZ \\ 2160=40WZ \\ divide\text{ both sides by 40} \\ \frac{2160}{40}=\frac{40WZ}{40} \\ 54=WZ \end{gathered}[/tex]Step 3
finally, find the perimeter of triangle XZW
Perimeter is the distance around the edge of a shape,so
[tex]Perimeter_{\Delta XZW}=XY+YZ+ZW+WV+VX[/tex]replace and calculate
[tex]\begin{gathered} Per\imaginaryI meter_{\Delta XZW}=XY+YZ+ZW+WV+VX \\ Perimeter_{\Delta XZW}=28+22.4+54+32+40 \\ Perimeter_{\Delta XZW}=176.4 \end{gathered}[/tex]therefore, the answer is
176.4
I hope this helps you
helpppppppppppppppppp
Answers
Answer:
Inverse should be:
f^(-1)(x) = -2x + 5
Step-by-step explanation:
A lighthouse beacon will illuminate to a distance of 12 km. If the lighthouse is located at (-5,2) on a grid, find the equation of the location of the furthest points lit the beacon.
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Light house is located at (-5,2)
Lighthouese beacon will illuminate a distnce =12 km
Use the distance formula to find the equation :
Distance formula is expressed as :
[tex]\begin{gathered} (x-a)^2+(y-b)^2=c^2 \\ \text{where (a,b) \& (x,y) are the coordinates and c is the distance} \end{gathered}[/tex]Substitute the given values :
[tex]undefined[/tex]The force of gravity is 6 times greater on the earth than it is on the moon. What is the weight of a 150-pound man on the moon?
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The force of gravity on the Earth is equal to 9.8m/s².
Now, if the force of gravity on the moon is 6 times lesser than Earth's gravity.
Then,
The weight of a 150-pound man on the moon is:
150-pound/ 6
= 25-pounds
Hence, the weight of the man is 25-pounds
48. In the parabola, y = 3x ^ 2 + 12x + 11 focus is located at a distance p > 0 from the vertex. Then p=a. 3b. 1/3c. 12d. 1/12e. None of the above
Answers
Given the equation,
[tex]y=3x^2+12x_{}+11[/tex]We are to solve for the vertex first, in order to solve for the vertex.
[tex]3x^2+12x+11=y[/tex]factor all through by 3
[tex]\begin{gathered} \frac{3x^2}{3}+\frac{12x}{3}+\frac{11}{3}=y \\ 3(x^2+4x+\frac{11}{3})=y\ldots\ldots.1 \end{gathered}[/tex][tex]x^2+4x=-\frac{11}{3}\text{ complete the square for the inner expression}[/tex][tex]\begin{gathered} x^2+4x+(\frac{4}{2})^2=-\frac{11}{3}+(\frac{4}{2})^2 \\ (x+2)^2=-\frac{11}{3}+4=\frac{1}{3} \\ =(x+2)^2-\frac{1}{3} \end{gathered}[/tex]Put (x+2)²-1/3 into equation 1
[tex]3((x+2)^2-\frac{1}{3})=y\ldots\ldots2[/tex]The vertex is at (-2,-1)
Note:
[tex]\begin{gathered} \text{vertex}=(h,k) \\ \text{focus}=(h,k+\frac{1}{4a}) \end{gathered}[/tex]P is the distance between the focus and the vertex.
[tex]\begin{gathered} (h-h,k+\frac{1}{4a}-k)=(0,\frac{1}{4a}) \\ \end{gathered}[/tex]where,
[tex]a=3\text{ from equation 2}[/tex]Therefore,
[tex]\begin{gathered} p=(0,\frac{1}{4\times3})=(0,\frac{1}{12}) \\ p=(0,\frac{1}{12}) \end{gathered}[/tex]Hence,
[tex]p=\frac{1}{12}[/tex]The correct answer is 1/12 [option D].
use the listing method to represent the following set. picture attached
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The correct option is A
{3, 4, 5, 6, ...}
Explanation:The condition given states that x is greater or equal to 3.
The only option that corresponds to this condition is:
{3, 4, 5, 6, ...}
The circumference of a circle is 278.71m. What is the approximate area of the circle? Use 3.14 for pi. Explain how the area of a circle changes when the circumference of a circle changes ( round the final answer to the nearest whole number as needed , round all the intermediate values to the nearest thousandth as needed )
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The circumference of a circle can be found through the formula:
[tex]C=2\cdot\pi\cdot r[/tex]clear the equation for the radius
[tex]r=\frac{C}{2\pi}[/tex]find the radius of the circumference
[tex]\begin{gathered} r=\frac{278.71}{2\pi} \\ r\approx44.358 \end{gathered}[/tex]find the area of the circle using the formula
[tex]\begin{gathered} A=\pi\cdot r^2 \\ A=\pi\cdot(44.358)^2 \\ A\approx6181 \end{gathered}[/tex]y=x2 shifted down 2 units and to the right 4 units
Answers
Answer:
y=(x-4)^2 -2
Step-by-step explanation:
the negative four means move to the right if it is positive it moves to the left in a graph
For each set of points below determine the distance between them using the distance formula. Express each answer in simplest radical form.
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Given data:
The given points are (5, 4) and (-1, 14).
The distance between the given points is,
[tex]\begin{gathered} d=\sqrt[]{(-1-5)^2+(14-4)^2} \\ =\sqrt[]{36+100} \\ =\sqrt[]{136} \\ =2\sqrt[]{34} \end{gathered}[/tex]Thus, the distance between the given points is 2√(34).
Evaluate the expression 10 to the 2 power + (3 +5 to the power 2) -5
The answer is 159
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The value of the expression 10 to the 2 power + (3 +5 to the power 2) -5 is 159.
What is an expression?An expression is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario.
The expression will be illustrated thus:
10² + (3 + 5)² - 5
= 100 + 8² - 5
= 100 + 64 - 5
= 164 - 5
= 159
The value is 159.
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Find the values of sin 0, cos 0, and tan e for the given right triangle. Give the exact values.sin 0=cos 0=tan 0=87
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We can use the definition:
[tex]\begin{gathered} \sin \theta=\frac{\text{opposite }}{\text{hypotenuse}} \\ \\ \cos \theta=\frac{\text{adjcent}}{\text{hypotenuse}} \\ \\ \tan \theta=\frac{\text{opposite}}{\text{adjacent}} \end{gathered}[/tex]Looking at the figure we can see the values:
But we don't have the hypotenuse value, we must use the Pythagorean theorem to find it
[tex]\begin{gathered} \text{hypotenuse = }\sqrt[]{7^2+8^2} \\ \\ \text{hypotenuse = }\sqrt[]{113} \end{gathered}[/tex]Now we have the hypotenuse we can find all values
[tex]\begin{gathered} \sin \theta=\frac{\text{opposite }}{\text{hypotenuse}}=\frac{8}{\sqrt[]{113}} \\ \\ \cos \theta=\frac{\text{adjcent}}{\text{hypotenuse}}=\frac{7}{\sqrt[]{113}} \\ \\ \tan \theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{7} \end{gathered}[/tex]9. Madison needs $10 000.00 in 16 years at an interest rate of 3 %/a compounded monthly. How much should she invest?
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SOLUTION:
Case: Compound interest
Method:
The formula is:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]P =?
A= $10 000.00
n = 12
r = 3% or 0.03
t = 16 years
[tex]\begin{gathered} 10000=P(1+\frac{0.03}{12})^{12\times16} \\ 10000=P(1.0025)^{192} \\ 10000=P\times1.6151 \\ P=\frac{10000}{1.6151} \\ P=6191.54 \end{gathered}[/tex]Final answer: To the nearest cent
She should invest $6191.54
Question 11 5 pts Find the value of x. Round to the nearest tenth. х 329 12. Not drawn to scale a. 10.2 b. 14.3 C. 10.4 d. 14.2
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Explanation
Step 1
Let
angle= 32
hypotenuse=x
adjacent side=12
so, we need a function that relates angel, hypotenuse and adjacent side
[tex]\text{cos}\emptyset=\frac{adjacent\text{ side}}{\text{hypotenuse}}[/tex]replace,
[tex]\begin{gathered} \text{cos}\emptyset=\frac{adjacent\text{ side}}{\text{hypotenuse}} \\ \text{cos32}=\frac{12}{\text{x}} \\ \text{Multiply both sides by x} \\ x\cdot\text{cos32}=\frac{12}{\text{x}}\cdot x \\ x\cdot\text{cos32}=12 \\ \text{divide both sides by cos 32} \\ \frac{x\cdot\text{cos32}}{\cos \text{ 32}}=\frac{12}{cos\text{ 32}} \\ x=14.15 \\ rounded \\ x=14.2 \end{gathered}[/tex]so, the answer is
[tex]d)x=14.2[/tex]I hope this helps you
A stock is worth $28,775 and drops 33% in one day. What percent does the stock have to grow the next day to get back to $28,775
Answers
ANSWER:
49.254%
STEP-BY-STEP EXPLANATION:
The first thing is to calculate the value after it has drops by 33%, like this:
[tex]\begin{gathered} 28775-28775\cdot33\% \\ \\ 28775-28775\cdot0.33 \\ \\ 28775-9495.75=19279.25 \end{gathered}[/tex]Now, we calculate what should grow by the following equation:
[tex]\begin{gathered} 19279.25+19279.25\cdot \:x=28775\: \\ \\ x=\frac{28775\:-19279.25}{19279.25} \\ \\ x=\frac{9495.75}{19279.25} \\ \\ x=0.49254\cong49.254\% \end{gathered}[/tex]The percent that should grow is 49.254%
Here’s the question. Just let me know when you have the answer. Just apart of a homework practice
Answers
By using the given zeros, we will see that the simplest polynomial is:
p(x) = x^3 - 7x - 6
So the correct option is the second one.
How to write the equation for the polynomial?Remember that the first simplest polynomial with the zeros x₁, x₂, x₃, ..., xₙ, is written as:
p(x) = (x - x₁)*(x - x₂)*...*(x - xₙ)
Here we have only 3 zeros, which are -1, -2, and 3, then we can write:
p(x) = (x - (-1))*(x - (-2))*(x - 3) = (x + 1)*(x + 2)*(x - 3)
Expanding the polynomial we get:
p(x) = (x + 1)*(x + 2)*(x - 3)
p(x) = (x^2 + x + 2x + 2)*(x - 3)
p(x) = (x^2 + 3x + 2)*(x - 3)
p(x) = x^3 + 3x^2 + 2x - 3x^2 - 9x - 6
p(x) = x^3 - 7x - 6
Then the correct option is the second one.
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how do I convert the rectangular equation: x=15 to a polar equation that expresses r in terms on theta?I got r=15 sectheta but wanted to double check
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In polar coordinates, the x variable is given as
[tex]x=r\cos \theta[/tex]So, we have the equation
[tex]r\cos \theta=15[/tex]By dividing both sides by cosine of thetat, we get
[tex]r=\frac{15}{\cos \theta}[/tex]since
[tex]\sec \theta=\frac{1}{\cos \theta}[/tex]The above result is equivalent to:
[tex]r=15\sec \theta[/tex]what is the the measure of each base angle of an isosceles triangle if it’s vertex angle measure is 44°?
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An isoceles triangle has one vertex angle and two congruent base angles, that is,
